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Derivatives of natural logs and exponents

  1. Nov 18, 2003 #1
    The problem I have is to find the derivative of the function:

    f(x) = (ln x)^x

    I know the derivative of ln x is 1/x, but the exponent is throwing me off. Can anyone offering any help? Thanks alot
     
  2. jcsd
  3. Nov 18, 2003 #2
    [tex]a^{b}=e^{b\ln a}[/tex]

    so [tex]\left( \ln x\right) ^{x}=e^{x\ln \left( \ln x\right) }[/tex]. now use product and chain rules.
     
    Last edited: Nov 18, 2003
  4. Nov 18, 2003 #3
    Another (very similar) approach is to take the log of both sides before you take the derivative.. use the chain rule to write d(ln(f(x))/dx in terms of df/dx, and solve for df/dx.
     
  5. Nov 18, 2003 #4
    Would I then have to substitute? I'm still not completely following... Thanks again
     
    Last edited: Nov 18, 2003
  6. Nov 18, 2003 #5
    as long as your answer has only [tex]x[/tex]'s in it, it should be ok. you could simplify the [tex]e^{x\ln \left( \ln x\right) }[/tex] back to [tex]\left( \ln x\right) ^{x}[/tex] if you want.
     
  7. Nov 18, 2003 #6
    would the derivative of (ln x)^x be:

    xe^(1/x) ??

    i don't know how to use the power and chain rule on e^xln(ln x)
     
  8. Nov 18, 2003 #7
    we have [tex]y=e^{x\ln \left( \ln x\right) }[/tex].

    this can be written as [tex]y=e^{u}[/tex] where [tex]u=x\ln \left( \ln x\right) [/tex].

    the chain rule is that [tex]\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}[/tex].

    [tex]\frac{dy}{du}=e^{u}[/tex]. to find [tex]\frac{du}{dx}[/tex], note that [tex]u[/tex] is the
    product of [tex]x[/tex] and [tex]\ln \left( \ln x\right) [/tex].

    [tex]\frac{du}{dx}=\left( \frac{d}{dx}x\right) \ln \left( \ln x\right) +x\frac{d
    }{dx}\ln \left( \ln x\right) [/tex]. [tex]\frac{d}{dx}x=1[/tex] and to find [tex]\frac{d}{dx}
    \ln \left( \ln x\right) [/tex], it may be useful to write [tex]v=\ln w[/tex] where [tex]w=\ln x
    [/tex].

    [tex]\frac{d}{dx}\ln \left( \ln x\right) =\frac{dv}{dx}=\frac{dv}{dw}\frac{dw}{dx
    }.[/tex]

    [tex]\frac{dv}{dw}=\frac{1}{w}[/tex] and [tex]\frac{dw}{dx}=\frac{1}{x}[/tex]. hence [tex]\frac{d
    }{dx}\ln \left( \ln x\right) =\frac{1}{w}\frac{1}{x}=\frac{1}{\ln x}\frac{1}{
    x}=\frac{1}{x\ln x}[/tex].

    putting this back into the most recent expression for [tex]\frac{du}{dx}[/tex], we
    get [tex]\frac{du}{dx}=1\cdot \ln \left( \ln x\right) +x\left( \frac{1}{x\ln x}
    \right) =\ln \left( \ln x\right) +\frac{1}{\ln x}[/tex].

    putting this back into the most recent expression for [tex]\frac{dy}{dx}[/tex], we
    get [tex]\frac{dy}{dx}=e^{u}\left( \ln \left( \ln x\right) +\frac{1}{\ln x}
    \right) =e^{x\ln \left( \ln x\right) }\left( \ln \left( \ln x\right) +\frac{1
    }{\ln x}\right) [/tex].

    since [tex]\left( \ln x\right) ^{x}=e^{x\ln \left( \ln x\right) }[/tex], we get [tex]
    \frac{dy}{dx}=e^{x\ln \left( \ln x\right) }\left( \ln \left( \ln x\right) +
    \frac{1}{\ln x}\right) =\left( \ln x\right) ^{x}\left( \ln \left( \ln
    x\right) +\frac{1}{\ln x}\right) [/tex]. either the middle or right side of this equation may be acceptable.
     
    Last edited: Nov 19, 2003
  9. Nov 18, 2003 #8
    I understand what you wrote, but I just can't figure out how you turned (ln x)^x into e^xln(ln x)

    Please elaborate. Otherwise, everything else has been very helpful.
     
  10. Nov 19, 2003 #9
    it's based on the property [tex]e^{\ln a}=a[/tex]. if we raise both sides to the [tex]b[/tex] power, we get [tex]\left( e^{\ln a}\right) ^{b}=a^{b}[/tex] which becomes [tex]a^{b}=e^{b\ln a}[/tex]. in this case, [tex]a=\ln x[/tex] and [tex]b=x[/tex].
     
    Last edited: Nov 19, 2003
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